Stabilization of Linear Systems with Structured Perturbations

The problem of stabilization of linear systems with bounded structured uncertainties are considered in this paper. Two notions of stability, denoted quadratic stability (Q-stability) and μ-stability, are considered, and corresponding notions of stabilizability and detectability are defined. In both cases, the output feedback stabilization problem is reduced via a separation argument to two simpler problems: full information (FI) and full control (FC). The set of all stabilizing controllers can be parametrized as a linear fractional transformation (LFT) on a free stable parameter. For Q-stability, stabilizability and detectability can in turn be characterized by Linear Matrix Inequalities (LMIs), and the FI and FC Q-stabilization problems can be solved using the corresponding LMIs. In the standard one-dimensional case the results in this paper reduce to well-known results on controller parametrization using state-space methods, although the development here relies more heavily on elegant LFT machinery and avoids the need for coprime factorizations

Ensemble Feedback Stabilization of Linear Systems

Stabilization of linear control systems with parameter-dependent system matrices is investigated. A Riccati based feedback mechanism is proposed and analyzed. It is constructed by means of an ensemble of parameters from a training set. This single feedback stabilizes all systems of the training set and also systems in its vicinity. Moreover its suboptimality with respect to optimal feedback for each single parameter from the training set can be quantified

A System Level Approach to Controller Synthesis

Biological and advanced cyber-physical control systems often have limited, sparse, uncertain, and distributed communication and computing in addition to sensing and actuation. Fortunately, the corresponding plants and performance requirements are also sparse and structured, and this must be exploited to make constrained controller design feasible and tractable. We introduce a new “system level” (SL) approach involving three complementary SL elements. SL parameterizations (SLPs) provide an alternative to the Youla parameterization of all stabilizing controllers and the responses they achieve, and combine with SL constraints (SLCs) to parameterize the largest known class of constrained stabilizing controllers that admit a convex characterization, generalizing quadratic invariance. SLPs also lead to a generalization of detectability and stabilizability, suggesting the existence of a rich separation structure, that when combined with SLCs is naturally applicable to structurally constrained controllers and systems. We further provide a catalog of useful SLCs, most importantly including sparsity, delay, and locality constraints on both communication and computing internal to the controller, and external system performance. Finally, we formulate SL synthesis problems, which define the broadest known class of constrained optimal control problems that can be solved using convex programming

On differential-algebraic control systems

In der vorliegenden Dissertation werden differential-algebraische Gleichungen (differential-algebraic equations, DAEs) der Form \ddt E x = Ax + f betrachtet, wobei $E$ und $A$ beliebige Matrizen sind. Falls $E$ nichtverschwindende Einträge hat, dann kommen in der Gleichung Ableitungen der entsprechenden Komponenten von $x$ vor. Falls $E$ eine Nullzeile hat, dann kommen in der entsprechenden Gleichung keine Ableitungen vor und sie ist rein algebraisch. Daher werden Gleichungen vom Typ \ddt E x = Ax + f differential-algebraische Gleichungen genannt. Ein Ziel dieser Dissertation ist es, eine strukturelle Zerlegung einer DAE in vier Teile herzuleiten: einen ODE-Anteil, einen nilpotenten Anteil, einen unterbestimmten Anteil und einen überbestimmten Anteil. Jeder Anteil beschreibt ein anderes Lösungsverhalten in Hinblick auf Existenz und Eindeutigkeit von Lösungen für eine vorgegebene Inhomogenität $f$ und Konsistenzbedingungen an $f$. Die Zerlegung, namentlich die quasi-Kronecker Form (QKF), verallgemeinert die wohlbekannte Kronecker-Normalform und behebt einige ihrer Nachteile. Die QKF wird ausgenutzt, um verschiedene Konzepte der Kontrollierbarkeit und Stabilisierbarkeit für DAEs mit~$f=Bu$ zu studieren. Hier bezeichnet $u$ den Eingang des differential-algebraischen Systems. Es werden Zerlegungen unter System- und Feedback-Äquivalenz, sowie die Folgen einer Behavioral-Steuerung $K_x x + K_u u = 0$ für die Stabilisierung des Systems untersucht. Falls für das DAE-System zusätzlich eine Ausgangs-Gleichung $y=Cx$ gegeben ist, dann lässt sich das Konzept der Nulldynamik wie folgt definieren: die Nulldynamik ist, grob gesagt, die Dynamik, die am Ausgang nicht sichtbar ist, d.h. die Menge aller Lösungs-Trajektorien $(x,u,y)$ mit $y=0$. Für rechts-invertierbare Systeme mit autonomer Nulldynamik wird eine Zerlegung hergeleitet, welche die Nulldynamik entkoppelt. Diese versetzt uns in die Lage, eine Behavior-Steuerung zu entwickeln, die das System stabilisiert, vorausgesetzt die Nulldynamik selbst ist stabil. Wir betrachten auch zwei Regelungs-Strategien, die von den Eigenschaften der oben genannten System-Klasse profitieren: Hochverstärkungs- und Funnel-Regelung. Ein System \ddt E x = Ax + Bu, $y=Cx$, hat die Hochverstärkungseigenschaft, wenn es durch die Anwendung der proportionalen Ausgangsrückführung $u=-ky$, mit $k>0$ hinreichend groß, stabilisiert werden kann. Wir beweisen, dass rechts-invertierbare Systeme mit asymptotisch stabiler Nulldynamik, die eine bestimmte Relativgrad-Annahme erfüllen, die Hochverstärkungseigenschaft haben. Während der Hochverstärkungs-Regler recht einfach ist, ist es jedoch a priori nicht bekannt, wie groß die Verstärkungskonstante $k$ gewählt werden muss. Dieses Problem wird durch den Funnel-Regler gelöst: durch die adaptive Justierung der Verstärkung über eine zeitabhängige Funktion $k(\cdot)$ und die Ausnutzung der Hochverstärkungseigenschaft wird erreicht, dass große Werte $k(t)$ nur dann angenommen werden, wenn sie nötig sind. Eine weitere wesentliche Eigenschaft ist, dass der Funnel-Regler das transiente Verhalten des Fehlers $e=y-y_{\rm ref}$ der Bahnverfolgung, wobei $y_{\rm ref}$ die Referenztrajektorie ist, beachtet. Für einen vordefinierten Performanz-Trichter (funnel) $\psi$ wird erreicht, dass $\|e(t)\|<\psi(t)$. Schließlich wird der Funnel-Regler auf die Klasse von MNA-Modellen von passiven elektrischen Schaltkreisen mit asymptotisch stabilen invarianten Nullstellen angewendet. Dies erfordert die Einschränkung der Menge der zulässigen Referenztrajektorien auf solche die, in gewisser Weise, die Kirchhoffschen Gesetze punktweise erfüllen.In this dissertation we study differential-algebraic equations (DAEs) of the form Ex'=Ax+f. One aim of the thesis is to derive the quasi-Kronecker form (QKF), which decomposes the DAE into four parts: the ODE part, nilpotent part, underdetermined part and overdetermined part. Each part describes a different solution behavior. The QKF is exploited to study the different controllability and stabilizability concepts for DAEs with f=Bu, where u is the input of the system. Feedback decompositions, behavioral control and stabilization are investigated. For DAE systems with output equation y=Cx, we may define the concept of zero dynamics, which are those dynamics that are not visible at the output. For right-invertible systems with autonomous zero dynamics a decomposition is derived, which decouples the zero dynamics of the system and allows for high-gain and funnel control. It is shown, that the funnel controller achieves tracking of a reference trajectory by the output signal with prescribed transient behavior. Finally, the funnel controller is applied to the class of MNA models of passive electrical circuits with asymptotically stable invariant zeros

Analysis And Control Of Networked Systems Using Structural And Measure-Theoretic Approaches

Network control theory provides a plethora of tools to analyze the behavior of dynamical processes taking place in complex networked systems. The pattern of interconnections among components affects the global behavior of the overall system. However, the analysis of the global behavior of large scale complex networked systems offers several major challenges. First of all, analyzing or characterizing the features of large-scale networked systems generally requires full knowledge of the parameters describing the system\u27s dynamics. However, in many applications, an exact quantitative description of the parameters of the system may not be available due to measurement errors and/or modeling uncertainties. Secondly, retrieving the whole structure of many real networks is very challenging due to both computation and security constraints. Therefore, an exact analysis of the global behavior of many real-world networks is practically unfeasible. Finally, the dynamics describing the interactions between components are often stochastic, which leads to difficulty in analyzing individual behaviors in the network. In this thesis, we provide solutions to tackle all the aforementioned challenges. In the first part of the thesis, we adopt graph-theoretic approaches to address the problem caused by inexact modeling and imprecise measurements. More specifically, we leverage the connection between algebra and graph theory to analyze various properties in linear structural systems. Using these results, we then design efficient graph-theoretic algorithms to tackle topology design problems in structural systems. In the second part of the thesis, we utilize measure-theoretic techniques to characterize global properties of a network using local structural information in the form of closed walks or subgraph counts. These methods are based on recent results in real algebraic geometry that relates semidefinite programming to the multidimensional moment problem. We leverage this connection to analyze stochastic networked spreading processes and characterize safety in nonlinear dynamical systems

From data and structure to models and controllers

Systems and control theory deals with analyzing dynamical systems and shaping their behavior by means of control. Dynamical systems are widespread, and control theory therefore has numerous applications ranging from the control of aircraft and spacecraft to chemical process control. During the last decades, a series of remarkable new control techniques have been developed. The majority of these techniques rely on mathematical models of the to-be-controlled system. However, the growing complexity of modern engineering systems complicates mathematical modeling. In this thesis, we therefore propose new methods to analyze and control dynamical systems without relying on a given system model. Models are thereby replaced by two other ingredients, namely measured data and system structure. In the first part of the thesis, we consider the problem of data-driven control. This problem involves the development of controllers for a dynamical system, purely on the basis of data. We consider both stabilizing controllers, and controllers that minimize a given cost function. Secondly, we focus on networked systems. A networked system is a collection of interconnected dynamical subsystems. For this type of systems, our aim is to reconstruct the interactions between subsystems on the basis of data. Finally, we consider the problem of assessing controllability of a dynamical system using its structure. We provide conditions under which this is possible for a general class of structured systems